It is the mean of the weighted summation over a window of length k and wt are the weights. Usually, the sequence w is generated using a window function. Numpy has a number of window functions already implemented: bartlett, blackman, hamming, hanning and kaiser. So, let's plot some Kaiser windows varying the parameter beta:
import numpy import pylab beta = [2,4,16,32] pylab.figure() for b in beta: w = numpy.kaiser(101,b) pylab.plot(range(len(w)),w,label="beta = "+str(b)) pylab.xlabel('n') pylab.ylabel('W_K') pylab.legend() pylab.show()The graph would appear as follows:
And now, we can use the function convolve(...) to compute the convolution between a vector x and one of the Kaiser window we have seen above:
def smooth(x,beta): """ kaiser window smoothing """ window_len=11 # extending the data at beginning and at the end # to apply the window at the borders s = numpy.r_[x[window_len-1:0:-1],x,x[-1:-window_len:-1]] w = numpy.kaiser(window_len,beta) y = numpy.convolve(w/w.sum(),s,mode='valid') return y[5:len(y)-5]Let's test it on a random sequence:
# random data generation y = numpy.random.random(100)*100 for i in range(100): y[i]=y[i]+i**((150-i)/80.0) # modifies the trend # smoothing the data pylab.figure(1) pylab.plot(y,'-k',label="original signal",alpha=.3) for b in beta: yy = smooth(y,b) pylab.plot(yy,label="filtered (beta = "+str(b)+")") pylab.legend() pylab.show()The program would have an output similar to the following:
As we can see, the original sequence have been smoothed by the windows.